Question

Find the value of \(\sin [\cot^{-1} \sqrt{2} (\cos (\tan^{-1} 1))]\).

Hint:

Composition of inverse functions looks intimidating but is just a series of substitutions. Keep track of the output of one function as the input for the next.

Answer 1

Concept:
• Solve inverse trigonometric expressions from the innermost bracket outward.
• Standard values: \(\tan^{-1} 1 = \frac{\pi}{4}\) and \(\cos \frac{\pi}{4} = \frac{1}{\sqrt{2}}\).

Step 1: Solve the innermost function
Identify the value of \(\tan^{-1} 1\): \[ \tan^{-1} 1 = \frac{\pi}{4} \] Now the expression becomes: \[ \sin [\cot^{-1} \sqrt{2} (\cos (\frac{\pi}{4}))] \]

Step 2: Solve the next layer
Substitute the value of \(\cos \frac{\pi}{4}\): \[ \cos \frac{\pi}{4} = \frac{1}{\sqrt{2}} \] The expression simplifies to: \[ \sin [\cot^{-1} (\sqrt{2} \cdot \frac{1}{\sqrt{2}})] \] \[ \sin [\cot^{-1} (1)] \]

Step 3: Evaluate the inverse cotangent
Find the angle whose cotangent is \(1\): \[ \cot^{-1} 1 = \frac{\pi}{4} \] The expression is now: \[ \sin (\frac{\pi}{4}) \]

Step 4: Calculate the final sine value
\[ \sin \frac{\pi}{4} = \frac{1}{\sqrt{2}} \]